EMF induced in a loop crossing a wall of field: 22-Elec-A7 Electromagnetics, May 2017, Q7

Question 7: EMF induced in a loop crossing a wall of field (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A rigid horizontal loop moving at constant speed perpendicular to the face of a slab region inside which the field is uniform and tilted, and zero outside.

Given data
Quantity	Symbol	Value
Loop area / side	$A$, $s$	$1$ m$^{2}$, $1$ m × $1$ m
Loop orientation	—	horizontal, sides N–S and E–W
Velocity	$v$	$30$ m/s due north
Field magnitude in the wall	$B$	$1\times 10^{-4}$ T
Field direction in the wall	—	east and $45^\circ$ up
Extent of the field region along the path	$W$	$100$ m
Field outside the wall	—	zero

Find. The induced EMF as a function of time, plotted from before the loop reaches the field region until after it has left.

Figure 7.1 — plan view and the resulting EMF. Only the upward component of B threads a horizontal loop, and only a changing overlap produces an EMF.

Approach. Resolve $\mathbf{B}$ onto the loop normal, write the flux as the field's normal component times the overlap area, and differentiate: the overlap grows linearly while the loop enters, is constant while it is immersed, and shrinks linearly while it leaves.

Only the vertical component of B counts. The loop is horizontal, so its normal is vertical and the eastward part of $\mathbf{B}$ contributes no flux at all: $$B_z = B\sin 45^\circ = (1\times 10^{-4})(0.7071) = \boxed{70.7\ \mu\text{T}} .$$
Maximum flux. When the loop is entirely inside the wall, $$\Phi_{\max} = B_z A = (70.71\times 10^{-6})(1) = 70.7\ \mu\text{Wb},$$ and this value is reached as soon as the trailing side crosses the near face.
Entering: the overlap area grows. With the leading side a distance $x$ past the face, the immersed area is $A = s\,x$ and $dx/dt = v$, so $$\left|\mathcal{E}\right| = \left|\frac{d\Phi}{dt}\right| = B_z\, s\, v = (70.71\times 10^{-6})(1)(30) = \boxed{2.12\ \text{mV}} ,$$ constant while the loop straddles the face. That lasts one loop length of travel, $$t_{\text{ramp}} = \frac{s}{v} = \frac{1}{30} = 33.3\ \text{ms}.$$
Fully immersed: no EMF. Once inside, the flux is $\Phi_{\max}$ and constant, so $\mathcal{E} = 0$ even though the loop is moving quickly through a strong field. The loop stays fully immersed while it travels $W - s = 99$ m: $$t_{\text{flat}} = \frac{W-s}{v} = \frac{99}{30} = 3.300\ \text{s}.$$
Leaving: the mirror image. Crossing the far face reverses the sign of $d\Phi/dt$, giving $+2.12$ mV for another $33.3$ ms. The whole event occupies $$t_{\text{total}} = \frac{W+s}{v} = \frac{101}{30} = 3.367\ \text{s},$$ so the trace is a narrow negative pulse, a long dead interval, and a narrow positive pulse — two spikes $3.300$ s apart, as plotted above.
Sign and current direction. Taking the loop normal upward, the flux increases on entry, so by Lenz's law the induced current opposes it and circulates clockwise seen from above; on exit it reverses to counter-clockwise. Writing the entry pulse as $\mathcal{E} = -d\Phi/dt = -2.12$ mV and the exit pulse as $+2.12$ mV records that reversal.
Cross-check with the motional form. Evaluating $\oint(\mathbf{v}\times\mathbf{B})\cdot d\boldsymbol{\ell}$ instead: with $\mathbf{v}$ north and $\mathbf{B}$ tilted east-and-up, $\mathbf{v}\times\mathbf{B}$ has an eastward part of magnitude $v B_z = 2.12$ mV/m. It acts only along whichever east–west side is inside the field, so the loop integral is $vB_z s = 2.12$ mV — identical to the flux answer, and it also shows plainly why a fully immersed loop gives zero: both E–W sides then contribute equally and cancel.

Check: the 100 m dimension is read as the extent of the field region along the direction of travel (north–south), since that is the only reading for which the flux history — and hence the requested plot — is determined; the east–west face is taken to be wide enough that the 1 m loop is fully spanned. Reading the 100 m as the east–west width instead would leave the slab thickness unspecified and the problem unsolvable.

Final results
Quantity asked	Result
Flux-producing component of B	$B_z = B\sin 45^\circ = 70.7\ \mu$T
Peak flux	$\Phi_{\max} = 70.7\ \mu$Wb
EMF while entering	$-2.12$ mV for $33.3$ ms (current clockwise from above)
EMF while fully immersed	$0$ for $3.300$ s
EMF while leaving	$+2.12$ mV for $33.3$ ms (current counter-clockwise from above)
Total duration of the event	$(W+s)/v = 3.367$ s
Motional cross-check	$vB_z s = 2.12$ mV on the immersed E–W side

Topic. Faraday's law for a rigid loop moving through a bounded region of uniform field.

Key relations. $\mathcal{E} = -d\Phi/dt$ with $\Phi = \int \mathbf{B}\cdot d\mathbf{A}$; motional form $\mathcal{E} = \oint(\mathbf{v}\times\mathbf{B})\cdot d\boldsymbol{\ell}$; for a horizontal loop only $B_z$ contributes; ramp EMF $= B_z s v$.

Why this works. Flux responds to the field component along the loop normal and to the overlap between the loop and the field region, so a uniform field produces an EMF only at the boundaries. Inside the region the two opposite sides sit in identical field and their motional contributions cancel exactly, which is the same statement as $d\Phi/dt = 0$; the two descriptions must agree, and checking them against each other is the cheapest guard against a sign or factor error. The resulting waveform — two short opposite pulses separated by a long silence — is the signature of a bounded field region much wider than the loop.

Common pitfall. Using the full $10^{-4}$ T instead of its vertical component (a factor $\sqrt{2}$ too big), or expecting a steady EMF while the loop is inside the field because it is still moving. A third slip is taking the ramp time as $W/v$ rather than $s/v$: the ramp lasts only as long as the loop straddles a face.

Source. Hayt & Buck, Engineering Electromagnetics 9th ed. ch. 10 (Faraday's law, motional EMF); Sadiku, Elements of Electromagnetics 7th ed. ch. 9.

Question 7 of 8: EMF induced in a loop crossing a wall of field

Question 7: EMF induced in a loop crossing a wall of field (20 marks)